next up previous contents
Next: Extended Kalman Filter Up: Kalman Filtering Technique Previous: Kalman Filtering Technique

Standard Kalman Filter

  When tex2html_wrap_inline3178 is a linear function


and we are able to write down explicitly a linear relationship


from tex2html_wrap_inline3180 , then the standard Kalman filter is directly applicable.


Figure 3: Kalman filter block diagram

Figure 3 is a block diagram for the Kalman filter. At time tex2html_wrap_inline3224 , the system model inherently in the filter structure generates tex2html_wrap_inline3226 , the best prediction of the state, using the previous state estimate tex2html_wrap_inline3228 . The previous state covariance matrix tex2html_wrap_inline3230 is extrapolated to the predicted state covariance matrix tex2html_wrap_inline3232 . tex2html_wrap_inline3232 is then used to compute the Kalman gain matrix tex2html_wrap_inline3236 and to update the covariance matrix tex2html_wrap_inline3238 . The system model generates also tex2html_wrap_inline3240 which is the best prediction of what the measurement at time tex2html_wrap_inline3224 will be. The real measurement tex2html_wrap_inline2849 is then read in, and the measurement residual  (also called innovation)


is computed. Finally, the residual tex2html_wrap_inline3248 is weighted by the Kalman gain matrix tex2html_wrap_inline3236 to generate a correction term and is added to tex2html_wrap_inline3226 to obtain the updated state tex2html_wrap_inline3254 .

The Kalman filter gives a linear, unbiased, and minimum error variance recursive algorithm to optimally estimate the unknown state of a linear dynamic system from noisy data taken at discrete real-time intervals. Without entering into the theoretical justification of the Kalman filter, for which the reader is referred to many existing books such as [8, 14], we insist here on the point that the Kalman filter yields at tex2html_wrap_inline3224 an optimal estimate of tex2html_wrap_inline3258 , optimal in the sense that the spread of the estimate-error probability density is minimized. In other words, the estimate tex2html_wrap_inline3254 given by the Kalman filter minimizes the following cost function


where M is an arbitrary, positive semidefinite matrix. The optimal estimate tex2html_wrap_inline3254 of the state vector tex2html_wrap_inline3258 is easily understood to be a least-squares estimate of  tex2html_wrap_inline3258 with the properties that [4]:

  1. the transformation that yields tex2html_wrap_inline3254 from tex2html_wrap_inline3274 is linear,
  2. tex2html_wrap_inline3254 is unbiased in the sense that tex2html_wrap_inline3278 ,
  3. it yields a minimum variance estimate with the inverse of covariance matrix of measurement as the optimal weight.

By inspecting the Kalman filter equations, the behavior of the filter agrees with our intuition. First, let us look at the Kalman gain  tex2html_wrap_inline3236 . After some matrix manipulation, we express the gain matrix in the form:


Thus, the gain matrix is ``proportional'' to the uncertainty in the estimate and ``inversely proportional'' to that in the measurement. If the measurement is very uncertain and the state estimate is relatively precise, then the residual tex2html_wrap_inline3248 is resulted mainly by the noise and little change in the state estimate should be made. On the other hand, if the uncertainty in the measurement is small and that in the state estimate is big, then the residual tex2html_wrap_inline3248 contains considerable information about errors in the state estimate and strong correction should be made to the state estimate. All these are exactly reflected in (20).

Now, let us examine the covariance matrix tex2html_wrap_inline3238 of the state estimate. By inverting tex2html_wrap_inline3238 and replacing tex2html_wrap_inline3236 by its explicit form (20), we obtain:


From this equation, we observe that if a measurement is very uncertain ( tex2html_wrap_inline3167 is big), the covariance matrix tex2html_wrap_inline3238 will decrease only a little if this measurement is used. That is, the measurement contributes little to reducing the estimation error. On the other hand, if a measurement is very precise ( tex2html_wrap_inline3167 is small), the covariance tex2html_wrap_inline3238 will decrease considerably. This is logic. As described in the previous paragraph, such measurement contributes considerably to reducing the estimation error. Note that Equation (21) should not be used when measurements are noise free because tex2html_wrap_inline3300 is not defined.

next up previous contents
Next: Extended Kalman Filter Up: Kalman Filtering Technique Previous: Kalman Filtering Technique

Zhengyou Zhang
Thu Feb 8 11:42:20 MET 1996